Probability of success (PoS): The PoS of a prospective clinical trial with \(N\)  subjects and prior information specified in Eq. (REF ) can be expressed as (e.g. see [1]})
\(\Phi \left( \frac{\sqrt{N}\cdot (\Delta _0 - \Delta _1) - r \cdot \tilde{\sigma } \cdot \gamma }{\sqrt{N\cdot \sigma _0^2 + r^2 \cdot \tilde{\sigma }^2}} \right) \) 
